Finally, in a unique application, \(L^1({\mathbb {R}}^d)\)  and \(H^\alpha ({\mathbb {R}}^d)\)  polynomial and exponential weighted propagation estimates have been very instrumental to developed error estimates for conservative spectral schemes for both the numerical error, consistency and stability to numerical solutions of the homogeneous Boltzmann flow by and Tharkabhushaman and authors [1]}, and also by Pennie and one of the authors for the homogeneous Landau flows [2]}.
